4.2 Two-Body Problem: N-N Scattering at Low Energies …
45
The expression for the amplitude shows a pole at k = iκ = i/a which corresponds
to a real or virtual bound state depending on whether the scattering length has a
positive or negative sign with binding energy, BE = 1/(ma
2
).
From the point of view of two nucleon systems at low energies, it is worth recalling
that the binding energy BE of deuteron is about 2.2 MeV, which corresponds to the
deuteron binding momentum, ℵ ≈ 45 MeV, which is smaller than pion mass by a
factor of about 3 indicating that the two nucleons are effectively at a distance three
times larger than the range of the nuclear force. In the case of spin singlet state
1 S 0 , the situation is even more dramatic: with negative BE ≈ −0.07 MeV which
corresponds to the momentum, ℵ ≈ 8 MeV, which is 20 times smaller than the pion
mass. Thus, deuteron as well as the virtual spin singlet s-state characterized by large
negative scattering length prove to be the realistic simplest examples of the weakly
bound two-body systems which can be studied by effective field theory using contact
interaction.
From the expression (4.5) for the scattering length, it becomes clear that the
scattering length diverges when C 0s is tuned to the value −2π
2
/(m). This behavior
is peculiar and illustrates an important basic principle of effective theories. Normally,
non-analytic behavior in long-distance observables arises from the analytic structure
at short distance parameters. Here the short-distance parameter is the strength of
the two-body contact interaction while the long-distance observable is the scattering
length. This only shows that divergence in a is essentially arising due to the quantum
fluctuations involving virtual particles with wave numbers less than where the
particles with wave numbers greater than are excluded by this cutoff. It should
also be noticed from the expression (4.7) that the coupling parameter C 0s diverges as
→ π/(2a). The physical observables are, however, independent of the parameter,
. It is therefore obvious that the effects of arbitrary large coupling constant have
to be compensated by equally large effects of quantum fluctuations involving virtual
particles.
From the point of view of the renormalization group, we start renormalizing
the expression (4.7) by multiplying both sides by m/(2π
2
) and redefining the
dimensionless coupling constant as
C 0s ≡
mC 0s
2π 2 . Thus, we re-express (4.7) as
C 0s =
2a/π
1 − 2a/π
= −
a
a − π/2
(4.11)
in terms of the coupling constant
C 0s . We note that as is varied keeping, a, fixed, the
plot of
C 0s as a function of maps a trajectory, called a renormalization group (RG)
trajectory. As is increased, approaching infinity, the constant
C 0s in Eq. (4.11)
approaches an ultraviolet fixed point, i.e.,
C 0s (() → −1 as → ∞
(4.12)
Alternatively, we may express the renormalization group as a differential equation
45
The expression for the amplitude shows a pole at k = iκ = i/a which corresponds
to a real or virtual bound state depending on whether the scattering length has a
positive or negative sign with binding energy, BE = 1/(ma
2
).
From the point of view of two nucleon systems at low energies, it is worth recalling
that the binding energy BE of deuteron is about 2.2 MeV, which corresponds to the
deuteron binding momentum, ℵ ≈ 45 MeV, which is smaller than pion mass by a
factor of about 3 indicating that the two nucleons are effectively at a distance three
times larger than the range of the nuclear force. In the case of spin singlet state
1 S 0 , the situation is even more dramatic: with negative BE ≈ −0.07 MeV which
corresponds to the momentum, ℵ ≈ 8 MeV, which is 20 times smaller than the pion
mass. Thus, deuteron as well as the virtual spin singlet s-state characterized by large
negative scattering length prove to be the realistic simplest examples of the weakly
bound two-body systems which can be studied by effective field theory using contact
interaction.
From the expression (4.5) for the scattering length, it becomes clear that the
scattering length diverges when C 0s is tuned to the value −2π
2
/(m). This behavior
is peculiar and illustrates an important basic principle of effective theories. Normally,
non-analytic behavior in long-distance observables arises from the analytic structure
at short distance parameters. Here the short-distance parameter is the strength of
the two-body contact interaction while the long-distance observable is the scattering
length. This only shows that divergence in a is essentially arising due to the quantum
fluctuations involving virtual particles with wave numbers less than where the
particles with wave numbers greater than are excluded by this cutoff. It should
also be noticed from the expression (4.7) that the coupling parameter C 0s diverges as
→ π/(2a). The physical observables are, however, independent of the parameter,
. It is therefore obvious that the effects of arbitrary large coupling constant have
to be compensated by equally large effects of quantum fluctuations involving virtual
particles.
From the point of view of the renormalization group, we start renormalizing
the expression (4.7) by multiplying both sides by m/(2π
2
) and redefining the
dimensionless coupling constant as
C 0s ≡
mC 0s
2π 2 . Thus, we re-express (4.7) as
C 0s =
2a/π
1 − 2a/π
= −
a
a − π/2
(4.11)
in terms of the coupling constant
C 0s . We note that as is varied keeping, a, fixed, the
plot of
C 0s as a function of maps a trajectory, called a renormalization group (RG)
trajectory. As is increased, approaching infinity, the constant
C 0s in Eq. (4.11)
approaches an ultraviolet fixed point, i.e.,
C 0s (() → −1 as → ∞
(4.12)
Alternatively, we may express the renormalization group as a differential equation